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The computer-generated reciprocal lattice of a fictional monoclinic 3D crystal. A two-dimensional crystal and its reciprocal lattice. Reciprocal lattice is a concept associated with solids with translational symmetry which plays a major role in many areas such as X-ray and electron diffraction as well as the energies of electrons in a solid.
Examples of determining indices for a plane using intercepts with axes; left (111), right (221) There are two equivalent ways to define the meaning of the Miller indices: [1] via a point in the reciprocal lattice, or as the inverse intercepts along the lattice vectors.
In a one-dimensional lattice the number of reciprocal lattice vectors that determine the bands in an energy interval is limited to two when the energy rises. In two and three dimensional lattices the number of reciprocal lattice vectors that determine the free electron bands E n ( k ) {\displaystyle E_{n}(\mathbf {k} )} increases more rapidly ...
The translation vectors define the nodes of Bravais lattice. The lengths of principal axes/edges, of unit cell and angles between them are lattice constants, also called lattice parameters or cell parameters. The symmetry properties of crystal are described by the concept of space groups. [1]
Another helpful ingredient in the proof is the reciprocal lattice vectors. These are three vectors b 1, b 2, b 3 (with units of inverse length), with the property that a i · b i = 2π, but a i · b j = 0 when i ≠ j. (For the formula for b i, see reciprocal lattice vector.)
The reciprocal lattices (dots) and corresponding first Brillouin zones of (a) square lattice and (b) hexagonal lattice. In mathematics and solid state physics , the first Brillouin zone (named after Léon Brillouin ) is a uniquely defined primitive cell in reciprocal space .
Ray diagram of Von Laue formulation. In physics, a Bragg plane is a plane in reciprocal space which bisects a reciprocal lattice vector, , at right angles. [1] The Bragg plane is defined as part of the Von Laue condition for diffraction peaks in x-ray diffraction crystallography.
The honeycomb point set is a special case of the hexagonal lattice with a two-atom basis. [1] The centers of the hexagons of a honeycomb form a hexagonal lattice, and the honeycomb point set can be seen as the union of two offset hexagonal lattices. In nature, carbon atoms of the two-dimensional material graphene are arranged in a honeycomb ...